
theorem
  for G being strict finite commutative Group st card G > 1 holds
  ex I be non empty finite set,
  F be associative Group-like commutative multMagma-Family of I st
  I = support (prime_factorization card G)
  & (for p be Element of I holds F.p is strict Subgroup of G &
  card (F.p) = (prime_factorization card G).p) &
  (for p,q be Element of I st p <> q holds
  (the carrier of (F.p)) /\ (the carrier of (F.q)) = {1_G})
  &
  (for y be Element of G
    ex x be (the carrier of G)-valued total I -defined Function
  st (for p be Element of I holds x.p in F.p) & y = Product x)
  &
  for x1,x2 be (the carrier of G)-valued total I -defined Function st
  (for p be Element of I holds x1.p in F.p) &
  (for p be Element of I holds x2.p in F.p) &
  Product x1 = Product x2 holds x1=x2
  proof
    let G be strict finite commutative Group;
    assume card G > 1; then
    consider I be non empty finite set,
    F be associative Group-like commutative multMagma-Family of I,
    HFG be Homomorphism of product F,G such that
    A1:
    I = support (prime_factorization card G)
    & (for p be Element of I holds F.p is strict Subgroup of G &
    card (F.p) = (prime_factorization card G).p) &
    (for p,q be Element of I st p <> q holds
    (the carrier of (F.p)) /\ (the carrier of (F.q)) ={1_G}) &
    HFG is bijective &
    for x be (the carrier of G)-valued total I -defined Function
    st for p be Element of I holds x.p in F.p
    holds x in product F & HFG.x =Product x by Th34;
    A2:for y be Element of G
    holds ex x be (the carrier of G)-valued total I -defined Function
    st (for p be Element of I holds x.p in F.p)
    & y = Product x
    proof
      let y be Element of G;
      y in the carrier of G;
      then y in rng HFG by A1,FUNCT_2:def 3;
      then
      consider x be object such that
      A3: x in the carrier of product F & y = HFG.x by FUNCT_2:11;
      reconsider x as total I-defined Function by A3,Lm6;
      A4: for p be Element of I holds x.p in F.p
      proof
        let p be Element of I;
        consider R be non empty multMagma such that
        A5: R = F.p & x.p in the carrier of R by Lm7,A3;
        thus x.p in (F.p) by A5;
      end;
      rng x c= the carrier of G
      proof
        let y be object;
        assume y in rng x; then
        consider i be object such that
        A6: i in dom x & y=x.i by FUNCT_1:def 3;
        reconsider i as Element of I by A6;
        consider R be non empty multMagma such that
        A7: R = F.i & x.i in the carrier of R by A3,Lm7;
        F.i is strict Subgroup of G by A1; then
        the carrier of (F.i) c= the carrier of G by GROUP_2:def 5;
        hence y in the carrier of G by A6,A7;
      end;
      then
      reconsider x as (the carrier of G)-valued
      total I -defined Function by RELAT_1:def 19;
      take x;
      thus thesis by A1,A3,A4;
    end;
    now let x1,x2 be (the carrier of G)-valued total I -defined Function;
      assume A8:
      (for p be Element of I holds x1.p in F.p) &
      (for p be Element of I holds x2.p in F.p) &
      Product x1 = Product x2;
      x1 in product F & HFG.x1 =Product x1 by A8,A1; then
      A9: HFG.x1 = HFG.x2 by A8,A1;
      x1 in the carrier of product F &
      x2 in the carrier of product F by A8,A1,STRUCT_0:def 5;
      hence x1=x2 by A9,A1,FUNCT_2:19;
    end;
    hence thesis by A1,A2;
  end;
